Their hopes rest on the GOP-controlled House passing its $659 million bill on Thursday. That vote is expected to be close — and could fail — but if it is approved, Reid and his leadership will be left with few paths forward with lawmakers itching to get out of town on Thursday afternoon.

The Senate’s own bill lacks unified Democratic support if it is not altered. And to avoid keeping lawmakers in town past Thursday, Reid will need the Senate GOP to agree to a sped-up vote series to deal with the Veterans Affairs reform legislation, the looming Highway Trust Fund insolvency and the border situation.

Republicans seem willing to grant a swift vote on border legislation if it’s the House’s pared-down border proposal, which delivers some emergency spending and GOP-backed policy changes to a 2008 anti-trafficking law that affords greater protections to young migrants from Central American countries. Democrats oppose this policy change, arguing it is inhumane, and believe the House’s bill shorts the Obama administration of resources.

Even Senate Republicans don’t love the House bill. But they want to pass something before turning off the lights on Capitol Hill for August recess — and before a large congressional delegation arrives to tour the Southern border at the end of the week.

“We don’t want to leave here with the message that we don’t support any kind of effort to take care of the emergency on the border,” said Sen. Dan Coats (R-Ind.). “The House bill is much closer in terms of what we think needs to be done immediately, that falls under the category of an emergency. Then there will be time to come back in September and address the longer term issue.”

While conservative GOP Sens. Ted Cruz of Texas and Jeff Sessions of Alabama have critiqued the House bill for not prohibiting President Barack Obama from expanding his Deferred Action for Childhood Arrivals policy, a broad spectrum of GOP senators are willing to support the House’s legislation even without that change. And the House may attach the DACA language on Thursday to attract conservative votes.

“By and large Republicans think it’s a positive step and a solution. There’s some who would go further for example on the DACA,” said Senate Minority Whip John Cornyn (R-Texas). “But I think the House has done good work.”

From conservative Oklahoma Sen. Jim Inhofe to deal-making South Carolina Sen. Lindsey Graham, Senate Republicans say they want to support the House’s border legislation before the Aug. 1 recess. It’s more palatable to them than Democratic leaders’ clean spending package that includes $2.7 billion for the border as well as $225 million for Israel and $615 million for wildfires.

While GOP senators like Cruz and Cornyn have their own proposals, they are unlikely to get votes on them in the Senate. At a closed-door party strategy meeting on Wednesday, Sen. Mike Johanns (R-Neb.) stood up to ask: Why don’t we back the House bill? Heads of conservatives and centrists nodded in agreement around the room, according to a source briefed on the meeting.

“The best approach is the House bill,” Johanns said in an interview after the GOP lunch. “That’s the common sense way of dealing with this. If Harry Reid would just put that on the floor, I believe there would be a sufficient number of Republicans to pass it and then we’d have the problem fixed before we head out for this August work period.”

Democrats, of course, have their own ideas. They have threatened to try to attach the Senate Gang of Eight comprehensive immigration reform bill to whatever the House passes and even offer the sweeping immigration bill as an amendment to the border supplemental bill.

But they are unlikely to get much Senate Republican support for those proposals or the underlying $3.57 billion spending package — which even some Senate Democrats oppose.

“The Democrats are in a bad spot. They have no chance of getting all their Democrats,” Graham said. “So you leave here, if you don’t do anything, with a bipartisan solution in front of you and the Obama solution rejected in a bipartisan fashion. That will be the political dynamic, if the House can pass it.”

Graham estimated that 90 percent of Senate Republicans could support the House bill.

Others weren’t so sure.

“Would all the Republicans go for the House bill? I don’t think so,” said one Senate Republican. “A majority might, but that doesn’t mean anything. It appears to me that nobody has any way forward here.”